Budget Constraint

A budget constraint is a line or plane that limits possible consumer choices.

Bivariate

A budget contraint in terms of two goods, x and y, is characterized as a line. The line is defined by the equality xP_x + yP_y = m; wherein

• i is the quantity of good i itself

• P_i is the price of good i

• m is the available income that can be spent

The intercepts of this line are m/P_x and m/P_y (on the x- and y-axes respectively).

The slope of this line is -P_x/P_y. This is generally referred to as the marginal rate of transformation (MRT).

Multivariate

A budget constraint is defined by the equality of some income constant m to the dot product of the price and quantity vectors p and q. (Recall that the dot product of two vectors like [x y] and [P_x P_y] is xP_x + yP_y.)

Also, the budget constraint is orthogonal to the price vector.

Non-linear Budget Sets

The above forms of budget constraints are linear. In some cases, the constraint is not constructed in this shape.

• Under autarky, the budget constraint conforms to the PPF.

• Rationing creates a piecewise budget constraint; it operates as normal up to some threshold where it becomes constant.
• Progressive (and regressive) taxing create complex piecewise budget constraints.

  • e.g., Abe's work on means-tested transfers

Usage

In neoclassical consumer choice theory, the optimal strategy is to pick the bundle where the budget constraint is tangent to the indifference curve:

m_budget = m_indif

MRT_xy = -P_x/P_y = -MU_x/MU_y

P_x/P_y = MU_x/MU_y

MU_x/P_x = MU_y/P_y

The intuitive explanation is that the optimal choice exists where, for all goods, the marginal utility given price is equal, so there is no incentive to substitute.

CategoryRicottone